Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 310924, 13 pages
doi:10.1155/2008/310924
Research Article
Nonlinear Boundary Value Problem for Concave Capillary Surfaces Occurring in Single Crystal Rod Growth from the Melt
1Department of Computer Science, Faculty of Mathematics and Computer Science, West University of Timisoara, 4, Vasile Parvan Boulevard, 300223 Timisoara, Romania
2Faculty of Physics, West University of Timisoara, 4, Vasile Parvan Boulevard, 300223 Timisoara, Romania
Received 6 May 2008; Revised 9 June 2008; Accepted 16 October 2008
Academic Editor: Michel Chipot
Copyright © 2008 Stefan Balint and Agneta Maria Balint. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The boundary value problem z″=((ρ⋅g⋅z−p)/γ)[1+(z′)2]3/2−(1/r)⋅[1+(z′)2]⋅z′, r∈[r1, r0], z′(r1)=−tan(π/2−αg), z′(r0)=−tanαc, z(r0)=0, and
z(r) is strictly decreasing on [r1,r0], is considered. Here, 0<r1<r0, ρ, g, γ, p, αc, αg are constants having the following properties: ρ, g, γ are strictly positive and 0<π/2−αg<αc<π/2. Necessary or sufficient conditions are given in terms of p for the existence of concave solutions of the above nonlinear boundary value problem (NLBVP). Numerical illustration is given. This kind of results is useful in the experiment planning and technology design of single crystal rod growth from the melt by edge-defined film-fed growth (EFG) method. With this aim, this study was undertaken.